Returns Distribution Analyzer: Examples
Short samples produce noisy higher moments; these scenarios are designed to show the direction of the change rather than any precise decimal. Skew measures asymmetry, excess kurtosis measures tail fatness relative to normal, and the Jarque-Bera statistic combines both into a normality test: a high statistic with a low p-value means non-normal returns. Compare what happens to these statistics as the series shifts from symmetric to skewed, and from thin-tailed to fat-tailed.
Worked Examples
See the inputs and outcome together
Each scenario keeps the starting point, the outcome, and the actual lesson in one place so the page reads like a decision notebook, not a data dump.
- 1
Symmetric series looks normal
A balanced series of paired gains and losses around a zero mean. No asymmetry and no fat tails.
Mean 0, skew 0, excess kurtosis minus 1.65, Jarque-Bera 1.36 (p = 0.51).
Returns
+/- 0.01, +/- 0.015, +/- 0.005, +/- 0.012, +/- 0.008, +/- 0.003
Bins
7
Skew is exactly zero because the series is symmetric, and the Jarque-Bera p-value of 0.51 fails to reject normality. The negative excess kurtosis reflects a flat, platykurtic shape from the small evenly-spaced sample, not a tail problem. This is the clean baseline.
- 2
Same calm, one crash day
Nine small moves around zero plus a single minus-8-percent shock. The kind of tail event that defines real return series.
Skew minus 2.19, excess kurtosis 3.32, Jarque-Bera 12.60 (p = 0.002).
Returns
0.005, 0.004, -0.003, 0.006, -0.002, 0.005, 0.003, -0.004, 0.005, -0.08
Bins
7
One crash day drives skew to minus 2.19 and excess kurtosis to 3.32, and Jarque-Bera now rejects normality at p = 0.002. The median is still positive, so a naive look at central tendency would miss the risk entirely. Higher moments, not the average, expose tail danger.
- 3
Mildly negative-skew series
A series with frequent small gains and a few larger losses, the typical fingerprint of a short-volatility or carry strategy.
Mean 0.002, skew minus 0.31, excess kurtosis minus 1.57, Jarque-Bera 1.19 (p = 0.55).
Returns
0.01, -0.02, 0.015, -0.005, 0.02, -0.01, 0.008, 0.012, -0.015, 0.005
Bins
5
Skew is mildly negative at minus 0.31, hinting at asymmetric losses, but with only ten observations Jarque-Bera cannot reject normality (p = 0.55). The lesson is statistical power: a real negative-skew profile may need dozens of observations before the normality test can flag it.
Patterns
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Sources & References
- A Test for Normality of Observations and Regression Residuals — Jarque, C. M. and Bera, A. K., International Statistical Review (1987)
- The Variation of Certain Speculative Prices — Mandelbrot, B., Journal of Business (1963)
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